Mister Exam

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Solving integrals/ (sin(x)dx)/(sqrt(2+cos(x)))

Integral of (sin(x)dx)/(sqrt(2+cos(x))) dx

The solution

You have entered [src]

 pi                  
  /                  
 |                   
 |      sin(x)       
 |  -------------- dx
 |    ____________   
 |  \/ 2 + cos(x)    
 |                   
/                    
pi                   
--                   
2

$$\int\limits_{\frac{\pi}{2}}^{\pi} \frac{\sin{\left(x \right)}}{\sqrt{\cos{\left(x \right)} + 2}}\, dx$$

Integral(sin(x)/sqrt(2 + cos(x)), (x, pi/2, pi))

Detail solution

Let $u = \sqrt{\cos{\left(x \right)} + 2}$ .

Then let $du = - \frac{\sin{\left(x \right)} dx}{2 \sqrt{\cos{\left(x \right)} + 2}}$ and substitute $- 2 du$ :

$\int \left(-2\right)\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\text{False}$
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int 1\, du = u$
  So, the result is: $- 2 u$
Now substitute $u$ back in:

$- 2 \sqrt{\cos{\left(x \right)} + 2}$
Add the constant of integration:

$- 2 \sqrt{\cos{\left(x \right)} + 2}+ \mathrm{constant}$

The answer is:

$- 2 \sqrt{\cos{\left(x \right)} + 2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                        
 |                                         
 |     sin(x)                  ____________
 | -------------- dx = C - 2*\/ 2 + cos(x) 
 |   ____________                          
 | \/ 2 + cos(x)                           
 |                                         
/

$$\int \frac{\sin{\left(x \right)}}{\sqrt{\cos{\left(x \right)} + 2}}\, dx = C - 2 \sqrt{\cos{\left(x \right)} + 2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

         ___
-2 + 2*\/ 2

$$-2 + 2 \sqrt{2}$$

         ___
-2 + 2*\/ 2

$$-2 + 2 \sqrt{2}$$

-2 + 2*sqrt(2)

Numerical answer [src]

0.82842712474619

0.82842712474619

Use the examples entering the upper and lower limits of integration.

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Identical expressions

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Integral of (sin(x)dx)/(sqrt(2+cos(x))) dx

Limits of integration:

The graph:

Piecewise:

The solution